in the statutes of ICC there is nothing said about completion of investigation. so i use Indian law to answer this question. my jurisdiction is India and i use its law so that i can give a direction from well established procedure of india which will work actually.

1. how final is the completion ?

final should mean a suspension of investigation and the revival of the investigation would take place if the pre-trial chamber deems fit.

here in India as per section 173 (8) crpc " Nothing in this section shall be deemed to preclude further investigation in respect of an offence after a report under sub- section (2) has been forwarded to the Magistrate and, where upon such investigation, the officer in charge of the police station obtains further evidence, oral or documentary, he shall forward to the Magistrate a further report or reports regarding such evidence in the form prescribed;"

1. what marks the end of investigation ?

the findings of satisfactory answer to all the items of final report that should be submitted after investigation, is the mark of completion of investigation.

according to section 173(2) crpc , the police must submit a final report before magistrate after completion of investigation. in that report the following items are mentioned.
(a) the names of the parties;

(b) the nature of the information;

(c) the names of the persons who appear to be acquainted with the circumstances of the case;

(d) whether any offence appears to have been committed and, if so, by whom;

(e) whether the accused has been arrested;

(f) whether he has been released on his bond and, if so, weather with or without sureties;

(g) whether he has been forwarded in custody under section 170.

(ii) The officer shall also communicate, In such manner as may be prescribed by the State Government, the action taken by him, to the person, if any, by whom the information relating to the commission of the offence was first given.

https://indiankanoon.org/doc/461024/

now as long as all the above mentioned items are not clearly determined, investigation is not thought to be complete. so as long as you cannot submit an answer to all these items of final report, your investigation is not complete.

1. should the prosecutor file a final report ?

yes the prosecutor should file a final report and submit it to pre-trial chamber.

in India according to Section 173(2), CrPC (criminal procedure code), the police submit a final report after completion of investigation.this final report to be submitted before magistrate. this marks the end of investigation.

in its judgement the learned madrass high court of india directed that "Therefore, I set aside the order of the trial court convicting the accused, as it cannot be held to be a legal order and I direct the Superintendent of Police, Sivaganga District to appoint some other Deputy Superintendent of Police to conduct fresh investigation on the complaint of the petitioner dated 10.11.1997 under his direct supervision and after finishing the investigation, the Deputy Superintendent of Police shall file a report before the learned Judicial Magistrate Devakottai for the appropriate offences within there months from the date of communication of this order. "

P. Alagarsamy vs State Of Tamil Nadu Rep. By ... on 23 September, 1999